if ` y= ( tan^(tan x)x)^(tan x) ` then at ` x=(pi)/( 4) , (dy)/(dx)`=

Options

A. 0
B. 1
C. 2
D. None

Answer

To solve the problem, we need to find the derivative of the function \( y = (\tan^{\tan x} x)^{\tan x} \) at \( x = \frac{\pi}{4} \). Let's go through the steps systematically.

### Step 1: Rewrite the Function
We start with the function:
\[ y = (\tan^{\tan x} x)^{\tan x} \]
Using the property of exponents \( (a^b)^c = a^{bc} \), we can rewrite \( y \) as:
\[ y = \tan^{\tan x \cdot \tan x} x = \tan^{\tan^2 x} x \]

### Step 2: Take the Natural Logarithm
To differentiate \( y \), we take the natural logarithm of both sides:
\[ \ln y = \tan^2 x \cdot \ln(\tan x) \]

### Step 3: Differentiate Both Sides
Now, we differentiate both sides with respect to \( x \):
Using the product rule on the right side:
\[ \frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx} \]
For the right side, we apply the product rule:
\[ \frac{d}{dx}(\tan^2 x \cdot \ln(\tan x)) = \frac{d}{dx}(\tan^2 x) \cdot \ln(\tan x) + \tan^2 x \cdot \frac{d}{dx}(\ln(\tan x)) \]

Calculating each derivative:
- The derivative of \( \tan^2 x \) is \( 2\tan x \sec^2 x \).
- The derivative of \( \ln(\tan x) \) is \( \frac{1}{\tan x} \cdot \sec^2 x \).

Putting it all together:
\[ \frac{1}{y} \frac{dy}{dx} = 2\tan x \sec^2 x \ln(\tan x) + \tan^2 x \cdot \frac{\sec^2 x}{\tan x} \]
This simplifies to:
\[ \frac{1}{y} \frac{dy}{dx} = 2\tan x \sec^2 x \ln(\tan x) + \tan x \sec^2 x \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Now, multiply both sides by \( y \):
\[ \frac{dy}{dx} = y \left( 2\tan x \sec^2 x \ln(\tan x) + \tan x \sec^2 x \right) \]

### Step 5: Substitute \( x = \frac{\pi}{4} \)
Now we need to evaluate \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \):
- At \( x = \frac{\pi}{4} \), we have \( \tan\left(\frac{\pi}{4}\right) = 1 \).
- Thus, \( y = \tan^{\tan^2\left(\frac{\pi}{4}\right)}\left(\frac{\pi}{4}\right) = 1^{1} = 1 \).

Now substituting into our derivative:
\[ \frac{dy}{dx} = 1 \left( 2 \cdot 1 \cdot 2 \cdot \ln(1) + 1 \cdot 2 \right) \]
Since \( \ln(1) = 0 \):
\[ \frac{dy}{dx} = 1 \left( 0 + 2 \right) = 2 \]

### Final Answer
Thus, the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is:
\[ \frac{dy}{dx} = 2 \]